Section4 Module7 Lagrangian Multipliers (1)

8/13/2019 Section4 Module7 Lagrangian Multipliers (1)

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that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunity

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Dynamic Simulation:

Lagrangian Multipliers

Objective

The objective of this module is to introduce Lagrangian multipliers

that are used with Lagranges equation to find the equations that

control the motion of mechanical systems having constraints.

The matrix form of the equations used by computer programs such asAutodesk Inventors Dynamic Simulation are also presented.


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Basic Problem in Multi-body Dynamics

In the previous module (Module 6)

we developed Lagranges equation

and showed how it could be used to

determine the equations of simple

motion systems.

Lagranges Equation

The examples we considered were for systems in which there

were no constraints between the generalized coordinates.

The basic problem of multi-body dynamics is to systematically findand solve the equations of motion when there are constraints that

bodies in the system must satisfy.

0

ii q

L

q

L

dt

d

Section 4Dynamic Simulation

Module 7Lagrangian Multipliers

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Non-conservative Forces

The derivation of Lagranges equation in the previous module (Module6) considered only processes that store and release potential energy.

These processes are called conservative because they conserve energy.

Lagranges equation must be modified to accommodate non-

conservative processes that dissipate energy (i.e. friction, damping, andexternal forces).

A non-conservative force or moment acting on generalized coordinate

qiis denoted as Qi.

The more general form of Lagranges equation is

i

ii

Qq

L

q

L

dt

d



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Simple Pendulum

Simple

Pendulum

c.g.

X

Y

The pendulum shown in thefigure will be used as an

example throughout this

module.

The position of the pendulumis known at any instance of

time if the coordinates of the

c.g.,Xcg,Ycg,and the angle q

are known.

Xcg,Ycgand qare the

generalized coordinates.

xy

Xcg

Ycg



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Kinetic and Potential Energies

The kinetic energy (T) and potentialenergy (V) of the pendulum are

These equations also give the

kinetic and potential energy of the

unconstrained body flying through

the air.

There needs to be a way to include

the constraints to differentiate

between the two systems.

cg

cgcg

mgYV

mYXmIT

2222

1

2

1

2

1 qc.g.

X

Y

xy

Xcg

Ycg

Unconstrained

Body



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Constraint Equations

In addition to satisfyingLagranges equations of motion,

the pendulum must satisfy the

constraints that the

displacements at X1and Y1are

zero.

The constraint equations are

c.g.

X

Y

xy

Xcg

Ycg

X1,Y1

0cos2

0sin2

1

1

YY

XX

cg

cg

q

q



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The c.g. lies on they-

axis halfway along the

length .


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Lagrangian Multipliers

X1

Y1

X

Y

The kinetic energy is augmented by

adding the constraint equationsmultiplied by parameters called

Lagrangian Multipliers.

Note that since the constraint

equations are equal to zero, we have

not changed the magnitude of the

kinetic energy.

The Lagrangian multipliers are

treated like unknown generalized

coordinates.

What are the units of 1and 2?

12

11

222

cos2

sin2

2

1

2

1

2

1

yY

XX

YmXmIT

cg

cg

cgcg

ql

ql

q



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Governing Equations

Lagranges Equation

Lagrangian



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i

ii

Qq

L

q

L

dt

d

i

n

i

i VTL 1

cgcgcgcgcg mgYYYXXYmXmIL

1211

222 cos2

sin22

1

2

1

2

1qlqlq

In the following slides, Lagranges

equation will be used in a systematic

manner to determine the equations of

motion for the pendulum.

The governing equations that will be

used are shown here.

There are no non-conservative forces

acting on the system ( ).0iQ


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Equation for 1stGeneralized CoordinateSection 4Dynamic Simulation


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1211

222 cos2

sin22

121

21 qlqlq

i

ii

Q

q

L

q

L

dt

d

Lagranges Equation

Generalized Coordinates

25

14

3

2

1

l

l

q

q

q

qYq

Xq

cg

cg1

1

1

1

l

q

L

Xmq

L

dt

d

Xm

q

L

cg

cg

1stEquation

01lcgXm

Mathematical Steps


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Equation for 2ndGeneralized CoordinateSection 4Dynamic Simulation


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222 cos2

sin22

121

21 qlqlq

i

ii

Q

q

L

q

L

dt

d

Lagranges Equation


25

14

3

2

1

l

l

q

q

q

q

Yq

Xq

cg

cg mgq

L

Ymq

L

dt

d

Ymq

L

cg

cg

2

2

2

2

l

2ndEquation

02

mgYm cg l

Mathematical Steps


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Equation for 3rdGeneralized Coordinate


1211

222 cos2

sin22

121

21 qlqlq

i

ii

Q

q

L

q

L

dt

d

Lagranges Equation


25

14

3

2

1

l

l

q

q

q

q

Yq

Xq

cg

cg qlql

q

q

sin2

cos2

21

3

3

3

q

L

Iq

L

dt

d

Iq

L

3rdEquation

Mathematical Steps

0sin2

cos2

21 qlqlq I



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Equation for 4thGeneralized CoordinateSection 4Dynamic Simulation


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222 cos2

sin22

121

21 qlqlq

i

ii

Q

q

L

q

L

dt

d

Lagranges Equation


25

14

3

2

1

l

l

q

q

q

q

Yq

Xq

cg

cg1

4

4

4

sin2

0

0

XXq

L

q

L

dt

dq

L

cg

q

4thEquation

Mathematical Steps

0sin2

1 XXcg q


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Equation for 5thGeneralized Coordinate



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222 cos2

sin22

121

21 qlqlq

i

ii

Q

q

L

q

L

dt

d

Lagranges Equation


25

14

3

2

1

l

l

q

q

q

q

Yq

Xq

cg

cg1

5

5

5

cos2

0

0

YYq

L

q

L

dt

d

q

L

cg

q

5thEquation

Mathematical Steps

0cos2

1 YYcg q


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Summary of EquationsSection 4Dynamic Simulation


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01lcgXm

02 mgYm cg l

0sin2

cos2

21 qlqlq I

0sin2

1 XXcg q

0cos2

1 YYcg q

There are five unknown generalizedcoordinates including the two

Lagrangian Multipliers. There are

also five equations.

Three of the equations aredifferential equations.

Two of the equations are algebraic

equations.

Combined, they are a system of

differential-algebraic equations

(DAE).


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Free Body Diagram Approach

1

2

cgmX

cgmY

cgI

The application of Lagrangesequation yields the same

equations obtained by drawing a

free-body diagram.

Free Body Diagram with

Inertial Forces

mg



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Summation of Forces in the X-direction

Summation of Forces in the Y-direction

Summation of Moments about the c.g.

01lcgXm

02 mgYm cg l

0sin2

cos2

21 qlqlq I

q 2

qcos2

qsin2


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Physical Significance of Lagrangian Multipliers

Force required to impose

the constraint that X1is a

constant.

Newtons 2ndLaw in x-direction

Lagrangian Multipliers are simply the forces (moments) required to

enforce the constraints. In general, the Lagrangian Multipliers are a

function of time, because the forces (moments) required to enforce

the constraints vary with time (i.e. depend on the position of the

pendulum).



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01lcgXm


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Matrix Format

i

i

ii q

VQ

q

T

q

T

dt

d

The computer implementation of Lagranges equation isfacilitated by writing the equations in matrix format.

Separating the Lagrangian into kinetic and potential

energy terms enables Lagranges equation to be written as

In this format, the conservative and non-conservative

forces are lumped together on the right hand side of the

equation.



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Matrix Format

The kinetic energy augmented with Lagrangian Multipliers can bewritten in matrix format as

tqqMqT TT ,2

1 l

Column array containing generalized coordinatevelocities.

Column array containing the constraint equations

(refer to Module 3 in this section).

Column array containing the Lagrangian multipliers.

Matrix containing the mass and mass moments of

inertia associated with each generalized coordinate.

q

tq,

l M

Inertia Matrix

000

000

000

000

A

cg

A

A

I

m

m

M



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2011 AutodeskFreely licensed for use by educational institutions. Reuse and changes require a note indicatingthat content has been modified from the original, and must attribute source content to Autodesk.

www.autodesk.com/edcommunityEducation Community

Matrix Format

Another equation for accelerationwas obtained in Module 4 based

on kinematics and the constraint

equations.

Combining this equation with

Lagranges equation from the

previous slide yields:

qqj

i

l

Qq

q

qM

j

i

j

i

0

Matrix Form of Equations

This equation can be solved

to find the accelerations

and constraint forces at an

instant in time.

The accelerations must

then be integrated to find

the velocities and positions.



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Solution of Differential-Algebraic

Equations (DAE)

The solution of even the simplest system of DAE requirescomputer programs that employ predictor-corrector type

numerical integrators.

The Adams-Moulton method is an example of the type of

numerical method used.

Significant research has led to the development of efficient and

robust integrators that are found in commercial computer

programs that generate, assemble, and solve these equations.

Autodesk Inventors Dynamic Simulation environment is an

example of such software.



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Module Summary

This module showed how Lagrangian Multipliers are used inconjunction with Lagranges equation to obtain the equations that

control the motion of mechanical systems.

The method presented provides a systematic method that forms the

basis of mechanical simulation programs such as Autodesk InventorsDynamic Simulation environment.

The matrix format of the equations were presented to provide insight

into the computations performed by computer software.

The Jacobian and constraint kinematics developed in Module 4 of this

section are an important part of the matrix formulation.



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Section4 Module7 Lagrangian Multipliers (1)